They have classical music because they didn't have to pay for it for the show.

I imagine, if you want an in-character explanation (and not pretend it still exists in the background), it's entirely possible that much of it was lost in the various wars (e.g. the Eugenics War, WWIII, etc.), in no small part because it was encumbered by many of these same copyright laws, which rendered it difficult to archive or distribute it.